Simplify tan^-1[3sin2α/(5+3 cos⁡〖2 α〗 )] + tan-1[tan⁡α/4] , where -π/2<α<π/2.

Explanation

`tan^(-1)[(3sin2alpha)/(5+3cos2alpha)]+tan^(-1)[tanalpha/4]`

`tan^(-1)[(6tanalpha)/(8+2tan^2alpha)]+tan^(-1)(tanalpha/4)`

`(6tan^2alpha)/(4(8+2tan^2alpha))<1`

`tan^(-1)[((3tanalpha)/(4ttan^2alpha)+tanalpha/4)/(1-(3tanalpha)/(4+tan^2alpha)*tanalpha/4)]`

`tan^(-1)((12tanalpha+4tanalpha+tan^3alpha)/(16+tan^2alpha))`

`tan^(-1)((16tanalpha+tan^3alpha)/(16+tan^2alpha))`

`tan^(-1)((tanalpha(16+tan^2alpha))/(16tan^2alpha))`

`tan^(-1)tanalpha=alpha`.